Bee in a Revolving Restaurant!

If there were a fly or a bee hovering in a theoretical revolving restaurant, would it move with the revolution of the room or stay stationary relative to the outside. The restaurant is theoretical in the sense that everything in it revolves, it is completely sealed and it never stops. I get tempted to see what would happen to an altitude neutral helium balloon, but it's not a perfect analogue - of course!

Staff: Mentor

As SteamKing said, if the bee were hovering in a car moving at a constant speed, then it would not move relative to the car. On the other hand, if the car were accelerating, the bee would tend to move backwards within the cabin. The bee doesn't have the seat to force it to accelerate forwards.

In a revolving restaurant, everything is accelerating towards the axis by forces imposed by the floor. This is analogous to the accelerating car situation. If the bee tried to hover in this setting, he couldn't do it. He would tend to drift to the outer wall of the restaurant, since he would tend to move in a straight line. Of course, if he wanted to, he could maintain his position by flying, using the air to accelerate him inward.

Staff: Mentor

The bee would feel a force slightly larger than gravity, slightly deflected relative to the usual direction. It can stay at rest relative to the (rotating) air in the restaurant easily, and rotate together with the restaurant.

Even in a stationary restaurant, an altitude neutral balloon would wander around based on where the air vents are. In your theoretical rotating (sealed and fan free) restaurant, the air would rotate along with everything else. Thus the balloon would rotate with the restaurant. I suspect that the balloon would NOT make it's way to the outer wall, because the air pressure would be higher there.

With a bee on the other hand, you have to define what the bee wants to do and how it does it. If it navigates visually and wants to hover, it would have no problem staying still relative to the furniture in the restaurant. If it was interested in something outside, it could hover relative to an outside object just as easily.

Loving some of the complexities being proposed - the idea of the helium balloon NOT moving to the edge of the restaurant due to higher air pressure is interesting! But the main concensus (which happily backs up what I have always argued) is that because the air in the room would be rotating with the room, the bee would rotate along with it. It would not be anchored by gravity to its position relative to the outside. Time to bring this back to the dinner table. (Take that husband and son!). Triumph no 2 after the chickens in the lorry argument!

A helium balloon inside a car (one that ISN'T neutrally buoyant) floats to the inside of the turn because the air inside the car is being 'sloshed' to the outside, forcing the balloon to the inside.

The bee is similar but opposite. It tends to be thrown to the outside of the rotating room, regardless of how still the air is with respect to the room. The bee is heavier than the air so it will need to fly in toward the center of the rotating room to stay stationary.

If that's hard to visualize, consider the room rotating extremely quickly, like a centrifuge. The bee is an object that wants to go in a straight line and without a force on it to push it toward the center of the room, it will be thrown to the outer wall.

Note that a helium balloon gets 'pushed' to the center of the room because the air is being thrown outward for the same reason.

Suppose that the restaurant is 30 meters in radius and is rotating once every 47 minutes. That's 4pi2r/t2 = 1.4x10-4 meters/second2. About 1/70,000 of a gee.

That's at right angles to gravity. Use vector addition to standard gravity (9.80665 meters/second2) and you get a resultant of 9.8066500011 meters/second2 at an angle of about .0008 degrees from vertical.

I would that expect a bee maneuvering based on visual feedback would not even notice discrepancies far larger than this.

So the next question is: is there a point where the mass of a mysteriously floating object is affected enough by the earth's gravity to overcome the pressure of the rotating air in the restaurant? If this object can maintain its altitude (without affecting the environment), but otherwise still be subject to gravity, would GRAVITY affect its HORIZONTAL velocity?

Say a larger object (an apple/Harry Potter/an elephant - all equally dense/aerodynamic) apparated into the room (and that apparation itself was momentumless - i.e didn't create any momentum or carry any from the point of origin). Does the downward pull of gravity reduce the horizontal push of the air pressure and slow down the rotation of the object? They should fall at the same rate, right? So, would they travel the same distance before they hit the floor?

Staff: Mentor

So the next question is: is there a point where the mass of a mysteriously floating object is affected enough by the earth's gravity to overcome the pressure of the rotating air in the restaurant?

What does that question mean? The mass of an object does not change at all, its gravitational force (weight) changes.
If an object can maintain its altitude*, it has the same density as the air at that altitude.

*note that this altitude is rotated some tiny fraction of a degree relative to the ground, as jbriggs444 calculated

Something non-rotating appearing in the room would be accelerated in the direction of rotation, slowing the rotation of the room (unless you power a motor to keep the angular velocity constant). This is independent of the vertical motion.